The present technology relates generally to waveform digitizing instruments. More specifically, the technology relates to waveform digitizing instruments consisting of analog to digital converters (ADCs) which are used to sample and digitize an analog signal, converting the analog signal to a sequence of values whose magnitude represents the voltage of the signal and the time location in the sequence represents when the signal was sampled.
Waveform digitizing instruments such as oscilloscopes are used to acquire waveforms for analysis. Of particular interest in the acquisition of waveforms is the bandwidth of the instrument. The instrument must be capable of acquiring the desired frequency content of the incoming signal. In other words, it must have sufficient speed to acquire the signal. Generally, we speak about bandwidth as the measure of instrument speed, and as such, the speed of signals it can acquire. There are several generally accepted definitions of bandwidth. One is the frequency at which the magnitude response of the instrument is down three decibels (dB) from the zero frequency (DC) response. Another is the highest frequency that the instrument can acquire. We will use the latter definition here.
A related and important instrument characteristic is the sample rate. Theoretically speaking, the sample rate of the instrument must meet the Nyquist criteria in order to avoid a well-known effect called frequency aliasing, or simply aliasing. Frequency aliasing is problematic and it is generally desirable to avoid this effect. The Nyquist criteria specifies that in order to avoid aliasing, the sample rate must be greater than twice the highest frequency that can be acquired by the instrument. In high-end instruments where the bandwidth is very high, high sample rates are difficult to achieve and sometimes the Nyquist criteria is barely met with instruments reaching sample rates of only two and one-half to three times the bandwidth. The Nyquist criteria is an absolute minimum, and it is generally accepted that sample rates of around ten times the bandwidth increase the usability of the instrument from a measurement standpoint. This is because at these high sample rates, lines can be drawn or inferred between each of the sample points taken. In order to achieve high sample rates relative to the bandwidth, two techniques are employed.
The first technique is that of time-interleaving. Time-interleaving is a technique whereby multiple ADCs sample the same analog input signal, but each of the multiple ADCs sample the signal at different times. Usually these different times are different phases of a sample clock that is a divided version of the overall interleaved sample rate. As an example, if two ADCs sample at 5 GS/s (i.e. with a sample period of 200 ps), but the second ADC samples the waveform with a sample phase shifted by 100 ps from the first ADC, then the two acquisitions from each of the ADCs can be put back together to form a resultant 10 GS/s acquisition. This technique of time interleaving is quite expensive in power and resources.
The second technique is that of waveform interpolation as explained in P. Pupalaikis, “The relationship between discrete-frequency s-parameters and continuous-frequency responses,” in DesignCon, IEC, February 2012. Waveform interpolation involves techniques to mathematically create the points in between the actually acquired waveform points. The theory behind this technique is a result of meeting the Nyquist criteria which essentially states that if the sample rate is high enough, then all sample points in the waveform can be generated mathematically from the acquired waveform. Waveform interpolation is often offered in the channel vertical control menu of modern oscilloscopes as an optional waveform processing step. Most often, the configuration of this interpolation involves the user determining an upsample factor, where the upsample factor is the factor to multiply by the hardware sample rate. As an example, a 5 GS/s oscilloscope channel sampling with 1 GHz of bandwidth (i.e. at a sample rate to bandwidth ratio of five), might be configured to interpolate with an upsample factor of two to achieve a user sample rate of 10 GS/s.
When time-interleaving is employed, the sample rate of the individual, interleaved digitizers need not meet the Nyquist criteria to avoid aliasing. Usually, for waveform interpolation to be employed usefully, the interpolation is applied to a waveform whose sample rate does meet the Nyquist criteria. Unfortunately, in oscilloscopes, interpolation is allowed whether the interpolation would be useful or not from an aliasing standpoint, and since an upsample factor is employed, the final sample rate becomes a product of the hardware sample rate employed and this factor which often creates waveform processing situations with user sample rates extending beyond what is needed practically.
A combination of these techniques can be employed such that time-interleaving is employed to reach the Nyquist criteria and waveform interpolation is utilized to get the sample rate higher from there.
Unfortunately, often within oscilloscopes, ADCs are time-interleaved to achieve sample-rates beyond those that are required to make good measurements when the resources could be withheld to save power or utilized for other, more beneficial purposes.
So far, this discussion has focused on, bandwidth and sample rate which are so-called horizontal characteristics because they pertain to the time axis of the acquired waveform. Of similar interest are the vertical characteristics of the waveform digitizer. In other words, waveform digitizers sample a waveform in two ways. The first way has already been discussed: the discretization of the time at which samples are taken. Waveform digitizers also discretize the waveform vertically in the form of quantization. The quantization characteristic is most often stated in bits. This is because, based on the typical construction of ADCs, there are a number of digital codes, or possible numbers, that can exist within the vertical acquisition range of the oscilloscope. Using traditional oscilloscope terminology, an oscilloscope has traditionally displayed waveforms on a grid with eight major divisions vertically. The gain and, therefore vertical range, of the oscilloscope is usually expressed as the voltage per division (VDIV) where the signal might be acquired in a range of ±4·VDIV. For the traditional eight bit oscilloscope, there might be 28=256 digital codes in this range. This quantization causes small errors in the waveform acquired which shows up statistically as a form of noise called quantization noise. To reduce this noise, oscilloscopes have been introduced, usually at slower sample rates, with higher resolution. Higher resolution means higher bits which means more codes which means less quantization noise.
Just as time-interleaving can be utilized to increase sample rate, vertical-interleaving can be utilized to increase resolution as proposed in M. McTigue and P. Byrne, “An 8-gigasample-per-second, 8-bit data acquisition system for a sampling digital oscilloscope.” Hewlett-Packard Journal, pp. 11-23, October 1993. Vertical interleaving is a technique whereby multiple ADCs sample the same analog input signal simultaneously, but each of the multiple ADCs sample the signal with different vertical offset amounts. Usually these different offsets are in multiples of specific fractions of a code. As an example, if two 8-bit ADCs sample at 5 GS/s, but the second ADC samples the waveform offset by half a code from the first ADC, then the two acquisitions from each of the ADCs can be put back together to form a resultant 5 GS/s, 9-bit acquisition.
While the finite resolution of the ADC creates noise, there are other sources of noise in the oscilloscope channel as well, mostly due to the front-end amplifier. In general, all forms of noise whether created by quantization or other effects are equally bad. For this reason, a figure-of-merit was established to describe the quality of a waveform digitizer or ADC. This figure-of-merit is called effective number of bits (ENOB). ENOB are calculated according to the signal to noise ratio (SNR) (in dB) as:
  ENOB  =            SNR      -      1.76        6.02  It can be shown that if an ADC with a given number of bits is perfect in all ways other than the fact that it quantizes an analog signal with a given number of bits, the SNR impact due to quantization noise will be such that the ENOB will equal the number of bits. Other additional sources of noise will degrade the ENOB.
There are mathematical ways of theoretically improving resolution that are so commonly used that these methods also appear in the channel menu of modern oscilloscopes. One method is called enhanced resolution (ERES). Generally, ERES is specified with the number of bits improvement desired, and this bits improvement specified drives a Gaussian filter as described in B. Orwiler, Oscilloscope Vertical Amplifiers—Circuit Concepts, pp. 21-38. Tektronix, 1 ed., 1969. The theoretical bits improvement desired determines the impulse response of filters as shown in FIG. 5. As shown in FIG. 6, the theoretical bits improvement comes as a result of a magnitude response effect that changes the −3 dB bandwidth of the channel. Unfortunately, the theoretical improvements are often not met and the actual improvement is unknown to the oscilloscope user. Furthermore, because the ERES filtering is manually controlled, there are often situations with regard to desired or needed bandwidth by the user which would benefit but ERES is not employed, resulting in waveform acquisitions that are noisier than necessary.
High resolution, like high sample-rate, is difficult to achieve because doubling either sample-rate or resolution means either doubling of ADC resources or doubling the speed of the ADC. Both lead to higher power consumption, larger size, and higher expense. As such, ADCs are precious resources.